A322807 -id:A322807 - cp's OEIS frontend

A322805 Lexicographically earliest such sequence a that for all i, j, a(i) = a(j) => f(i) = f(j), where f(1)=0, f(2)=1, f(n)=-1 for odd primes, and f(n) = A252463(n) for any other number.

1, 2, 3, 4, 3, 5, 3, 6, 6, 7, 3, 8, 3, 9, 8, 10, 3, 11, 3, 12, 12, 13, 3, 14, 11, 15, 10, 16, 3, 17, 3, 18, 16, 19, 17, 20, 3, 21, 22, 23, 3, 24, 3, 22, 14, 25, 3, 26, 27, 27, 28, 28, 3, 29, 24, 30, 31, 32, 3, 33, 3, 34, 23, 35, 36, 36, 3, 31, 37, 38, 3, 39, 3, 40, 20, 37, 38, 41, 3, 42, 18, 43, 3, 44, 41, 45, 46, 47, 3, 48, 49, 46, 50, 51, 52, 53, 3, 54, 30, 55, 3, 52, 3, 56, 33

Offset: 1

Antti Karttunen, Dec 26 2018

nonn

For all i, j:
  a(i) = a(j) => A319694(i) = A319694(j),
  a(i) = a(j) => A319699(i) = A319699(j),
  a(i) = a(j) => A319700(i) = A319700(j),
  a(i) = a(j) => A319703(i) = A319703(j),
  a(i) = a(j) => A319989(i) = A319989(j),
  a(i) = a(j) => A320110(i) = A320110(j) => A320111(i) = A320111(j).

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A252463.
Cf. A319694, A319699, A319700, A319703, A319989, A320110, A320111.
Cf. also A322807, A322809.

up_to = 65537;
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};
A252463(n) = if(!(n%2),n/2,A064989(n));
A322805aux(n) = if(n<=2,n-1,if(isprime(n),-1,A252463(n)));
v322805 = rgs_transform(vector(up_to,n,A322805aux(n)));
A322805(n) = v322805[n];

A322809 Lexicographically earliest such sequence a that a(i) = a(j) => f(i) = f(j) for all i, j, where f(n) = -1 if n is an odd prime, and f(n) = floor(n/2) for all other numbers.

Original entry on oeis.org

1, 2, 3, 4, 3, 5, 3, 6, 6, 7, 3, 8, 3, 9, 9, 10, 3, 11, 3, 12, 12, 13, 3, 14, 14, 15, 15, 16, 3, 17, 3, 18, 18, 19, 19, 20, 3, 21, 21, 22, 3, 23, 3, 24, 24, 25, 3, 26, 26, 27, 27, 28, 3, 29, 29, 30, 30, 31, 3, 32, 3, 33, 33, 34, 34, 35, 3, 36, 36, 37, 3, 38, 3, 39, 39, 40, 40, 41, 3, 42, 42, 43, 3, 44, 44, 45, 45, 46, 3, 47, 47, 48, 48, 49, 49, 50, 3, 51, 51, 52, 3, 53, 3, 54, 54

Offset: 1

Antti Karttunen, Dec 26 2018

nonn

This sequence is a restricted growth sequence transform of a function f which is defined as f(n) = A004526(n), unless n is an odd prime, in which case f(n) = -1, which is a constant not in range of A004526. See the Crossrefs section for a list of similar sequences.
For all i, j:
  A305801(i) = A305801(j) => a(i) = a(j),
  a(i) = a(j) => A039636(i) = A039636(j).
For all i, j: a(i) = a(j) <=> A323161(i+1) = A323161(j+1).
The shifted version of this filter, A323161, has a remarkable ability to find many sequences related to primes and prime chains. - Antti Karttunen, Jan 06 2019

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences related to binary expansion of n

Cf. A004526, A039636, A305801, A323161.
A list of few similarly constructed sequences follows, where each sequence is an rgs-transform of such function f, for which the value of f(n) is the n-th term of the sequence whose A-number follows after a parenthesis, unless n is of the form ..., in which case f(n) is given a constant value outside of the range of that sequence:
  A322809 (A004526, unless an odd prime) [This sequence],
  A322589 (A007913, unless an odd prime),
  A322591 (A007947, unless an odd prime),
  A322805 (A252463, unless an odd prime),
  A323082 (A300840, unless an odd prime),
  A322822 (A300840, unless n > 2 and a Fermi-Dirac prime, A050376),
  A322988 (A322990, unless a prime power > 2),
  A323078 (A097246, unless an odd prime),
  A322808 (A097246, unless a squarefree number > 2),
  A322816 (A048675, unless an odd prime),
  A322807 (A285330, unless an odd prime),
  A322814 (A286621, unless an odd prime),
  A322824 (A242424, unless an odd prime),
  A322973 (A006370, unless an odd prime),
  A322974 (A049820, unless n > 1 and n is in A046642),
  A323079 (A060681, unless an odd prime),
  A322587 (A295887, unless an odd prime),
  A322588 (A291751, unless an odd prime),
  A322592 (A289625, unless an odd prime),
  A323369 (A323368, unless an odd prime),
  A323371 (A295886, unless an odd prime),
  A323374 (A323373, unless an odd prime),
  A323401 (A323372, unless an odd prime),
  A323405 (A323404, unless an odd prime).

up_to = 65537;
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
A322809aux(n) = if((n>2)&&isprime(n),-1,(n>>1));
v322809 = rgs_transform(vector(up_to,n,A322809aux(n)));
A322809(n) = v322809[n];

a(n) = A323161(n+1) - 1.

A322822 Lexicographically earliest such sequence a that a(i) = a(j) => f(i) = f(j) for all i, j, where f(2) = -1, f(n) = 0 if n is a Fermi-Dirac prime (A050376) > 2, and f(n) = A300840(n) for all other numbers.

Original entry on oeis.org

1, 2, 3, 3, 3, 4, 3, 5, 3, 6, 3, 7, 3, 8, 9, 3, 3, 10, 3, 11, 12, 13, 3, 7, 3, 14, 15, 16, 3, 9, 3, 17, 18, 19, 20, 21, 3, 22, 23, 11, 3, 12, 3, 24, 25, 26, 3, 27, 3, 28, 29, 30, 3, 15, 31, 16, 32, 33, 3, 34, 3, 35, 36, 37, 38, 18, 3, 39, 40, 20, 3, 21, 3, 41, 42, 43, 44, 23, 3, 45, 3, 46, 3, 47, 48, 49, 50, 24, 3, 25, 51, 52, 53, 54, 55, 27, 3, 56, 57, 58, 3, 29, 3, 30

Offset: 1

Antti Karttunen, Dec 29 2018

nonn

For all i, j: a(i) = a(j) => A322823(i) = A322823(j).

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A050376, A052330, A052331, A302777, A300840, A322823.

Cf. also A322805, A322807.

up_to = 65537;
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
ispow2(n) = (n && !bitand(n,n-1));
A302777(n) = ispow2(isprimepower(n));
A050376list(up_to) = { my(v=vector(up_to), i=0); for(n=1,oo,if(A302777(n), i++; v[i] = n); if(i == up_to,return(v))); };
v050376 = A050376list(up_to);
A050376(n) = v050376[n];
A052330(n) = { my(p=1,i=1); while(n>0, if(n%2, p *= A050376(i)); i++; n >>= 1); (p); };
A052331(n) = { my(s=0,e); while(n > 1, fordiv(n, d, if(((n/d)>1)&&ispow2(isprimepower(n/d)), e = vecsearch(v050376, n/d); if(!e, print("v050376 too short!"); return(1/0)); s += 2^(e-1); n = d; break))); (s); };
A300840(n) = A052330(A052331(n)>>1);
A322822aux(n) = if((2==n),-1,if(A302777(n),0,A300840(n)));
v322822 = rgs_transform(vector(up_to,n,A322822aux(n)));
A322822(n) = v322822[n];

A322806 Lexicographically earliest such sequence a that a(i) = a(j) => A285330(i) = A285330(j) for all i, j.

Original entry on oeis.org

1, 2, 3, 3, 4, 5, 6, 4, 5, 7, 8, 9, 10, 11, 9, 6, 12, 13, 14, 15, 15, 16, 17, 18, 7, 19, 11, 20, 21, 22, 23, 8, 18, 24, 13, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 22, 38, 39, 40, 41, 42, 29, 43, 44, 45, 46, 47, 48, 49, 50, 10, 37, 51, 52, 28, 53, 54, 55, 56, 57, 58, 59, 60, 25, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 38

Offset: 1

Antti Karttunen, Dec 26 2018

nonn

Restricted growth sequence transform of A285330.

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A048675, A285328, A285330, A285332, A285333, A286543, A322807.

up_to = 65537;
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
A007947(n) = factorback(factorint(n)[, 1]); \\ From A007947
A048675(n) = my(f = factor(n)); sum(k=1, #f~, f[k, 2]*2^primepi(f[k, 1]))/2; \\ From A048675
A285328(n) = { my(r); if((n > 1 && !bitand(n,(n-1))), (n/2), r=A007947(n); if(r==n,1,n = n-r; while(A007947(n) <> r, n = n-r); n)); };
A285330(n) = if(issquarefree(n),A048675(n),A285328(n));
v322806 = rgs_transform(vector(up_to,n,A285330(n)));
A322806(n) = v322806[n];

A322811 a(1) = 0; for n > 1, a(n) = A001221(A285330(n)).

Original entry on oeis.org

0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 2, 2, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 1, 2, 2, 2, 2, 1, 2, 2, 2, 1, 1, 1, 2, 2, 1, 1, 2, 1, 2, 3, 2, 1, 2, 2, 2, 3, 2, 1, 3, 1, 2, 2, 1, 2, 1, 1, 2, 3, 1, 1, 2, 1, 2, 2, 2, 2, 2, 1, 2, 1, 2, 1, 3, 2, 2, 2, 2, 1, 3, 2, 2, 3, 3, 3, 2, 1, 2, 2, 2, 1, 1, 1, 2, 2

Offset: 1

Antti Karttunen, Dec 27 2018

nonn

For all i, j:
  A322806(i) = A322806(j) => a(i) = a(j),
  A322807(i) = A322807(j) => a(i) = a(j).

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A001221, A048675, A285330, A322806, A322807, A322812.

A007947(n) = factorback(factorint(n)[, 1]); \\ From A007947
A048675(n) = my(f = factor(n)); sum(k=1, #f~, f[k, 2]*2^primepi(f[k, 1]))/2; \\ From A048675
A285328(n) = { my(r); if((n > 1 && !bitand(n,(n-1))), (n/2), r=A007947(n); if(r==n,1,n = n-r; while(A007947(n) <> r, n = n-r); n)); };
A285330(n) = if(issquarefree(n),A048675(n),A285328(n));
A322811(n) = if(1==n,0,omega(A285330(n)));

a(1) = 0; for n > 1, a(n) = A001221(A285330(n)).

If n > 1 is squarefree, a(n) = A322812(n) = A001221(A048675(n)), otherwise a(n) = A001221(A285328(n)) = A001221(n).

A322861 Lexicographically earliest such sequence a that a(i) = a(j) => A278222(A285330(i)) = A278222(A285330(j)) for all i, j.

Original entry on oeis.org

1, 2, 2, 2, 2, 3, 2, 2, 3, 4, 2, 3, 2, 4, 3, 2, 2, 3, 2, 4, 4, 4, 2, 4, 4, 4, 4, 5, 2, 5, 2, 2, 4, 4, 3, 3, 2, 4, 4, 4, 2, 6, 2, 6, 7, 4, 2, 4, 5, 4, 4, 6, 2, 3, 4, 5, 4, 4, 2, 7, 2, 4, 8, 2, 4, 6, 2, 4, 4, 6, 2, 9, 2, 4, 10, 6, 3, 6, 2, 6, 9, 4, 2, 8, 4, 4, 4, 6, 2, 7, 4, 11, 4, 4, 4, 4, 2, 5, 4, 4, 2, 6, 2, 6, 5

Offset: 1

Antti Karttunen, Dec 31 2018

nonn

Restricted growth sequence transform of A278222(A285330(n)).
For all i, j:
  A322806(i) = A322806(j) => a(i) = a(j),
  A322807(i) = A322807(j) => a(i) = a(j),
  a(i) = a(j) => A322862(i) = A322862(j).

Cf. A048675, A278222, A285330, A286622, A322806, A322807, A322862, A322868.

up_to = 65537;
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); t };
A007947(n) = factorback(factorint(n)[, 1]); \\ From A007947
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };  \\ From A046523
A048675(n) = my(f = factor(n)); sum(k=1, #f~, f[k, 2]*2^primepi(f[k, 1]))/2; \\ From A048675
A285328(n) = { my(r); if((n > 1 && !bitand(n,(n-1))), (n/2), r=A007947(n); if(r==n,1,n = n-r; while(A007947(n) <> r, n = n-r); n)); };
A285330(n) = if(moebius(n)<>0,A048675(n),A285328(n));
A278222(n) = A046523(A005940(1+n));
v322861 = rgs_transform(vector(up_to,n,A278222(A285330(n))));
A322861(n) = v322861[n];

cp's OEIS Frontend

A322805 Lexicographically earliest such sequence a that for all i, j, a(i) = a(j) => f(i) = f(j), where f(1)=0, f(2)=1, f(n)=-1 for odd primes, and f(n) = A252463(n) for any other number.

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A322809 Lexicographically earliest such sequence a that a(i) = a(j) => f(i) = f(j) for all i, j, where f(n) = -1 if n is an odd prime, and f(n) = floor(n/2) for all other numbers.

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A322822 Lexicographically earliest such sequence a that a(i) = a(j) => f(i) = f(j) for all i, j, where f(2) = -1, f(n) = 0 if n is a Fermi-Dirac prime (A050376) > 2, and f(n) = A300840(n) for all other numbers.

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A322806 Lexicographically earliest such sequence a that a(i) = a(j) => A285330(i) = A285330(j) for all i, j.

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A322811 a(1) = 0; for n > 1, a(n) = A001221(A285330(n)).

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A322861 Lexicographically earliest such sequence a that a(i) = a(j) => A278222(A285330(i)) = A278222(A285330(j)) for all i, j.

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